Semantics of First Order Parametric Specifications

Parametricity is one of the most effective ways to achieve compositionality and reuse in software development. Parametric specifications have been thoroughly analyzed in the algebraic setting and are by now a standard part of most software development toolkits. However, an effort towards classifying, specifying and refining algorithmic theories, rather than mere datatypes, quickly leads beyond the realm of algebra, and often to full first order theories. We extend the standard semantics of parametric specifications to this more general setting. The familiar semantic characterization of parametricity in the algebraic case is expressed in terms of the free functor, i.e. using the initial models. In the general case, initial models may not exist, and the free functor is not available. Various syntactic, semantic, and abstract definitions of parametricity have been offered, but their exact relationships are often unclear. Using the methods of categorical model theory, we establish the equivalence of two well known, yet so far unrelated, definitions of parametricity, one syntactic, one semantic. Besides providing support for both underlying views, and a way for aligning the systems based on each of them, the offered general analysis and its formalism open several avenues for future research and applications.